Exercise [02.03]

mikey · 28 Mar 2008, 09:48

OK,

Sorry if this is badly laid out. This is mostly to satisfy myself.

To show AB+BC=AC

where A, B and C are successive points along a hyperbolic straight line assuming that the general property of logarhythms is true $\log ab = \log a + \log b$ (1):

Using $AB=\log \frac {QA.PB}{QB.PA}$ and $BC=\log \frac {QB.PC}{QC.PB}$

$AB+BC=\log { \frac {QA.PB}{QB.PA} } + \log { \frac {QB.PC}{QC.PB} }$

$=\log { \frac {QA.PB}{QB.PA}.\frac {QB.PC}{QC.PB} }$ using (1)

$= \log { \frac {QA.PB.QB.PC}{QB.PA.QC.PB} }$

$=\log { \frac {QA.PC}{PA.QC} } = AC$

Exercise [02.03]

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